Ideas
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Guardian:
The odds that a kitsune would loose their hoshi no tama were small given how protective they were of such things, that another would find it and keep it without knowing what it was- that was even smaller. Kurama had the misfortune of something one of a kind happening- he lost his hoshi no tama, it was found and kept by a human whom didn't know what it was and it was absorbed into the human. Now he has to act as protector to the human so that he won't die do to the stupid worm's actions.
OR
The Kyuubi Attack turns out differently when Kurama winds up Kakashi's unwilling protector.
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Slender Man:
What if Kakashi saw Slender Man during his chūnin exam and wrote it off? Then is followed but not noticing- since paranoia is part of being a ninja- until Sakumo's death as he either sees it when he finds the body or at the funeral? Then snippets of him being stalked, steadily chipping at his sanity, until it cumulates to the Kyuubi attack when in he just gives in…
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Paralyzed:
While attempting to create Chidori, something goes wrong that causes the lightning to paralyze Kakashi. Minato ends up checking on him and, finding him unable to move [looking very much as if he were dead], takes him to the hospital. The medics attempt to use chakra to cure him but his chakra fights against it and thus creating a "stalemate" between them that worsens his condition and makes it appear as if he has no chakra thus declaring him dead.
Kakashi is then dissected, with Minato's hesitant approval, since he's an organ donor- he is awake and aware the entire time, dying only when his heart is removed as the medics take very good care of his body to keep his organs in the best shape.
Sad thing is, if a Hyūga or Uchiha or even an Aburame had visited, they would have noticed that he was alive- he would have come out of it on his own eventually also- but the only ones whom would have visited [here Hiashi, Shibi and Fugaku are Sakumo's genin team] were out on a mission at the time.
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Serial Killer AU:
Rin would do "mercy killings" or "euthanasia" for old people, terminally ill people, people in comas, etc., Obito would be trying to kill "bad guys" but at times relapse into simply killing out of revenge while pushing his wrongs onto a target, Minato will go for other killers but also rapists.
Kakashi will, mainly because he wants a challenge, go after organizations- gangs, illegal brothels, black markets, human trafficking, etc.- which would lead into the Danzo thing as Danzo will be running ROOT but they're basically assassins for hire whom generally target the world leaders in such a way to ignite wars so they'd get more business taking care of the enemy leaders (never mind that the enemy changes with which employer has the most money).
I see it as them having "territories" (loosely) with them overlapping a bit but in a way that Minato and Kakashi never met face-to-face but have heard about each other through Rin and Obito- I think the first time they really meet would be when Kakashi goes after Danzo and gets in trouble.
I also see them having "day jobs" (kinda like twisted superhero secret identities); Minato would be a teacher (either High School or College), Rin will be a nurse, Obito I think would be in civil services (maybe a fireman or policeman?) and Kakashi would be…Hm…a college student/waiter?
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Kamui Caused Crossover:
["Jarvis, what the hell is this frequency?"]
Kakashi blinked at the voice coming from his communicator, having never heard an accent quiet like that before, but another voice (Jarvis?) answered- with a different yet similarly unknown accent- the first.
{"I have cross-referenced all frequencies known to Earth but have been unable to find a match, Sir."}
It was, Kakashi decided as the men's voices started off on the possibility of it being a satellite frequency they had tapped into, time to make the two aware of him, "Yo."
OR
Kakashi decided to work his way up to larger things when training Kamui, the communicator he practiced gets reception from someone in an entirely different dimension- Tony Stark to be exact.
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Because I watched Pursuit of Happiness:
There's a little boy that always sits and 'barters' a story out of him in exchange for some of the kid's lunch every day that the elementary school down the street is open. He does not expect to save the boy from kidnappers- he certainly does not expect the father, grateful and kind, to offer him the spare bedroom upon finding out he's homeless.
But, then, when has anything ever gone how Kakashi expected it?
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Newspapers:
ANBU were nameless, faceless and emotionless things that would kill you at the drop of a coin. People feared them, were terrified of them- to the point where it harmed their duties. So, it was decided that there had to be a way for ANBU to seem more 'human' without giving their identities away. He didn't know who picked a newspaper- "A Newspaper By Unnamed Shinobi", ANBUs- but Kakashi believed they deserved a slow death.
[Or in which all ANBU are to write a piece for a newspaper that will be available to all villagers, Kakashi has no idea what to write and daydreams of revenge for the person who thought of the idea.]